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A329767 Triangle read by rows where T(n,k) is the number of binary words of length n >= 0 with runs-resistance k, 0 <= k <= n. 19
1, 2, 0, 0, 2, 2, 0, 2, 2, 4, 0, 2, 4, 6, 4, 0, 2, 2, 12, 12, 4, 0, 2, 6, 30, 18, 8, 0, 0, 2, 2, 44, 44, 32, 4, 0, 0, 2, 6, 82, 76, 74, 16, 0, 0, 0, 2, 4, 144, 138, 172, 52, 0, 0, 0, 0, 2, 6, 258, 248, 350, 156, 4, 0, 0, 0, 0, 2, 2, 426, 452, 734, 404, 28, 0, 0, 0, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
A composition of n is a finite sequence of positive integers with sum n.
For the operation of taking the sequence of run-lengths of a finite sequence, runs-resistance is defined as the number of applications required to reach a singleton.
Except for the k = 0 column and the n = 0 and n = 1 rows, this is the triangle appearing on page 3 of Lenormand, which is A319411. Unlike A318928, we do not here require that a(n) >= 1.
The n = 0 row is chosen to ensure that the row-sums are A000079, although the empty word arguably has indeterminate runs-resistance.
LINKS
Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003.
EXAMPLE
Triangle begins:
1
2 0
0 2 2
0 2 2 4
0 2 4 6 4
0 2 2 12 12 4
0 2 6 30 18 8 0
0 2 2 44 44 32 4 0
0 2 6 82 76 74 16 0 0
0 2 4 144 138 172 52 0 0 0
0 2 6 258 248 350 156 4 0 0 0
0 2 2 426 452 734 404 28 0 0 0 0
For example, row n = 4 counts the following words:
0000 0011 0001 0010
1111 0101 0110 0100
1010 0111 1011
1100 1000 1101
1001
1110
MATHEMATICA
runsres[q_]:=If[Length[q]==1, 0, Length[NestWhileList[Length/@Split[#]&, q, Length[#]>1&]]-1];
Table[Length[Select[Tuples[{0, 1}, n], runsres[#]==k&]], {n, 0, 10}, {k, 0, n}]
CROSSREFS
Row sums are A000079.
Column k = 2 is A319410.
Column k = 3 is 2 * A329745.
The version for compositions is A329744.
The version for partitions is A329746.
The number of nonzero entries in row n > 0 is A319412(n).
The runs-resistance of the binary expansion of n is A318928.
Sequence in context: A029361 A275966 A284059 * A356018 A107502 A230419
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Nov 21 2019
STATUS
approved

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Last modified July 25 02:27 EDT 2024. Contains 374585 sequences. (Running on oeis4.)